A Gale–Ryser type characterization of potentially -bigraphic pairs

Let A and B be nonincreasing lists of nonnegative integers, having lengths m and n , respectively. The pair ( A ; B ) is potentially K s , t -bigraphic if there is a simple bipartite graph containing K s , t (with s vertices in the part of size m and t in the part of size n ) such that the lists of vertex degrees in the two partite sets are A and B . We give a characterization of potentially K s , t -bigraphic pairs, generalizing the Gale–Ryser characterization of bigraphic pairs.